On Frobenius graphs of diameter 3 for finite groups
Thomas Breuer, László Héthelyi, Erzsébet Horváth, Burkhard Külshammer
本研究针对有限群的Frobenius图，探究其直径为3的成立条件，该图用于记录有限群不可约特征标限制到子群上的分量。
对于有限群$G$的子群$H$，Frobenius图$Γ(G, H)$记录$G$的不可约特征标限制到$H$上的分量。我们研究该图直径为3的条件。
For a subgroup $H$ of a finite group $G$, the Frobenius graph $Γ(G, H)$ records the constituents of the restrictions to $H$ of the irreducible characters of $G$. We investigate when this graph has diameter 3.
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Morgan and Parker have proved that if $G$ is a group satisfying the condition that $Z(G) = 1$, then the connected components of the commuting graph of $G$ have diameter at most $10$. Parker has proved that if in addition $G$ is solvable, then the commuting graph of $G$ is disconnected if and only if $G$ is a Frobenius group or a $2$-Frobenius group, and if the commuting graph of $G$ is connected, then its diameter is at most $8$. We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap Z(G) = 1$. We also prove that if $G$ is solvable and $G/Z(G)$ is either a Frob…
The cyclic graph of a group $G$ is the graph whose vertices are the nonidentity elements of $G$ and whose edges connect distinct elements $x$ and $y$ if and only if the subgroup $\langle x,y\rangle$ is cyclic. We obtain information about the cyclic graph of $2$-Frobenius groups. The cyclic graph of a $2$-Frobenius group is disconnected. In this paper, we determine the number of connected components of the cyclic graph of any $2$-Frobenius group.
A group $G$ has a Frobenius graphical representation (GFR) if there is a simple graph $\varGamma$ whose full automorphism group is isomorphic to $G$ and it acts on vertices as a Frobenius group. In particular, any group $G$ with GFR is a Frobenius group and $\varGamma$ is a Cayley graph. The existence of an infinite family of groups with GFR whose Frobenius kernel is a non-abelian $2$-group has been an open question. In this paper, we give a positive answer by showing that the Higman group $A(f,q_0)$ has a GFR for an infinite sequence of $f$ and $q_0$.
In this paper, we give a characterization for a class of edge-transitive Cayley graphs, and provide methods for constructing Cayley graphs with certain symmetry properties. Also this study leads to construct and characterise a new family of half-transitive graphs.
The prime graph of a finite group $G$ is denoted by $\ga(G)$ whose vertex set is $π(G)$ and two distinct primes $p$ and $q$ are adjacent in $\ga(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $\ga(H)=\ga(G)$, in while $H\not\cong G$. In this paper, we consider finite groups with the same prime graph as the almost simple group $\textrm{PGL}(2,49)$. Moreover, we construct some Frobenius groups whose their prime graph coincide with $\ga(\textrm{PGL}(2,49))$, in particular, we get that $\textrm{PGL}(2,49)$…
Normal Edge-Transitive Cayley Graphs of Frobenius Groups
一个群$G$的Cayley图称为正规边传递的，如果它允许$G$的全形（$\operatorname{Sym}(G)$中$G$的正则副本的正规化子）的某个子群的边传递作用.通过对两个素数的乘积为a阶的弗罗贝尼乌斯群的Cayley图的研究，完成了这类图的分类.我们确定这些图的自同构群，特别证明有一个独特的顶点本原的例子，即Fano平面的旗图。
The diameter of the commuting graph of a finite group with trivial centre
研究了具有平凡中心的有限群的交换图。结果表明，交换图的连通分支的直径最多为10。
The power graph of finite group G is a simple graph whose vertex set is G and two distinct elements a and b are adjacent if and only if one of them is a power of the other. The proper power graph of G is a graph which is obtained by deleting the identity vertex (the identity element of G). In this paper, we improve the diameter bound of proper power graph of alternating group of degree n which the graph is connected. We show that the diameter of An is between 6 and 11, if the n at least 51. We also describe a number of short paths in these power graphs.
Let \(G\) be a finite group, and let \(Δ(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(Δ(G)\) is connected, the diameter of \(Δ(G)\) is at most \(3\). In the present paper, we provide a description of the finite solvable groups for which the diameter of this graph attains the upper bound. This also enables us to confirm a couple of conjectures proposed by M.L. Lewis.
For a finite group $G$, let $Δ(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we show that if the diameter of $Δ(G)$ is equal to three, then the complement of $Δ(G)$ is bipartite. Also in this case, we determine the structure of the character graph $Δ(G)$.
On the diameter of intersection graphs of finite groups
本文回答了 Csákány 和 Pollák 关于有限群交图直径的两个长期未解问题，给出了非单群交图直径为 4 的必要条件，并首次构造了直径为 4 的非单群和交错群实例。
Matroids from gain graphs over quotient groups
本文提出了一种从商群上的增益图构造Matroid的新方法，证明了Frobenius群的必要性。
There is no upper bound for the diameter of the commuting graph of a finite group
构造了一类具有直径递增的交换图的有限特殊2-群
Elementary proofs of the diameter bounds for the power graphs
本文给出简化证明，表明增强幂图与幂图的补图在除去孤立顶点后连通且直径至多3。
On Bounding the Diameter of the Commuting Graph of a Group
群$G$的交换图是简单无向图，其顶点是$G$的非中心元素，且两个不同的顶点相邻当且仅当它们交换。Jafarzadeh和Iranmanesh证明了当交换图连通时，有限群的交换图的直径存在一个泛上界.在本文中，我们确定的上界的直径交换图的某些类别的群体，以排除他们作为可能的反例，这一猜想。我们也给出了一个例子，一个无限族的群平凡中心和直径为6，以前最大的已知直径为5的一个无限族的$S_n$。
Frobenius Problem and dead ends in integers
设a和b是正的、互素的整数。我们证明了以下等价性：（i）d是Z关于a和b的（对称）Cayley图中的死端，（ii）d是关于a和b的Frobenius值（它不能写成a和b的非负或非正整数线性组合），并且d是关于这个性质的极大值（在Cayley图中）.此外，对于给定的整数a和b，我们明确地描述了Z中所有这样的元素。最后，我们证明了Z关于任何有限的对称生成集只有2000个死端。在附录中，我们证明了每个有限生成群都有一个生成集，关于这个生成集存在死胡同。
Notes on the diameter of the complement of the power graph of a finite group
该研究确定了有限群幂图补图与增强幂图补图各连通分支的直径，完整解答了Peter J. Cameron提出的两个相关问题。
General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer $ H \leq \mathrm{GL}(V) $, where the vector space $ V $ has dimension $ d $ over the prime field, can be bounded in terms of $ d $ and $ \log |V| / \log |H| $ only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.
The diameter and dominating sets of the difference graph of a nilpotent group
本文研究了nilpotent群的差图的直径和支配集，证明其直径上界为4，并分类了所有直径为k的nilpotent群。
Let $G$ be a finite group, and $α$ a nontrivial character of $G$. The McKay graph ${\mathcal M}(G,α)$ has the irreducible characters of $G$ as vertices, with an edge from $χ_1$ to $χ_2$ if $χ_2$ is a constituent of $αχ_1$. We study the diameters of McKay graphs for simple groups $G$. For $G$ a group of Lie type, we show that for any $α$, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for $G={\rm PSL}_n(q)$ or ${\rm PSU}_n(q)$. We also bound the diameter for symmetric and alternating groups.
Let $Γ$ be a group and $(Γ_n)_{n=1} ^{\infty}$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $Γ/Γ_n$ , under some natural hypotheses on the behaviour of power and commutator words in $Γ$. The bounds we obtain do not depend on a choice of generating set. Moreover under reasonable conditions our method provides a fast algorithm for constructing directed Cayley graphs of diameter satisfying our bounds. The proof is closely analogous to the the Solovay-Kitaev procedure, which only uses commutator …
We prove that the number of iterations taken by the Weisfeiler-Leman algorithm for configurations coming from Schreier graphs is closely linked to the diameter of the graphs themselves: an upper bound is found for general Schreier graphs, and a lower bound holds for particular cases, such as for Schreier graphs with $G=\mbox{SL}_{n}({\mathbb F}_{q})$ ($q>2$) acting on $k$-tuples of vectors in ${\mathbb F}_{q}^{n}$; moreover, an exact expression is found in the case of Cayley graphs.
Diameters of commuting graphs of partial transformation semigroups
本文针对有限集上的部分变换半群，确定其交换图直径，证明其与已知的对应变换半群交换图直径相等，证实存在半群与其真子半群交换图直径相等的结论。
We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, …
Graph-theoretical estimates of the diameters of the Rubik's Cube groups
本文提出一种基于图阶数、度数、围长及过顶点圈数等局部参数的对称图直径严格下界方法，并应用于魔方群凯莱图，获得比随机正则图下界更紧、达到正确直径60%至77%的估计。
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